Molecular moments based on the deformation density

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 717

Section 8.7.3.4.1.2. Molecular moments based on the deformation density

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.3.4.1.2. Molecular moments based on the deformation density

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The moments derived from the total density ρ(r) and from the deformation density Δρ(r) are not identical. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as $[\eqalignno{ \mu _{xx}\left (\rho _{{\rm total}}\right) &=\textstyle\int \rho x^2{\,{\rm d}}{\bf r} \cr &=\textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r}+\int \Delta\rho x^2{\,{\rm d}}{\bf r.} & (8.7.3.34)}]$ The promolecule is the sum over spherical atom densities, or $[\eqalignno{ \textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r} &= \textstyle\int \textstyle\sum \limits _{{i}}\rho _{{\rm spherical\ atom,}i}x^2{\,{\rm d}}{\bf r} \cr &= \textstyle\sum \limits _{i} \textstyle\int \rho _{{\rm spherical\ atom,}i}x^2{\,{\rm d}}{\bf r.} & (8.7.3.35)}]$ If R_i = (X_i, Y_i, Z_i) is the position vector for atom i, each single-atom contribution can be rewritten as $[\eqalignno{ \mu _{{i},\,xx,\,{\rm spherical\ atom}} &= \textstyle\int \rho _{i{\rm,\, spherical\ atom}}x^2{\,{\rm d}}{\bf r} \cr &= \textstyle\int \rho _{i{\rm,\, spherical\ atom}} (x-X_{i}) ^2{\,{\rm d}}{\bf r} \cr &\quad+X_{i} \textstyle\int \rho _{i{\rm,\, spherical\ atom}}2(x-X_{{i}}) {\,{\rm d}}{\bf r} \cr &\quad+X_{i}^2 \textstyle\int \rho _{i{\rm,\, spherical\ atom}}{\,{\rm d}}{\bf r.}\ & (8.7.3.36)}]$ Since the last two integrals are proportional to the atomic dipole moment and its net charge, respectively, they will be zero for neutral spherical atoms. Substitution in (8.7.3.35) gives, with $[\langle(x-X) {_{i}^2}\rangle ={1\over3}\langle r_{{i}}^2\rangle]$ , and $[\langle r_{i}^2\rangle =\int \rho _{i}(r) r^2{\,{\rm d}}{\bf r},]$ $[ \textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r}= \textstyle{1\over3}\textstyle\sum \limits _{{\rm atoms}}\langle r^2\rangle _{{\rm spherical\ atom}},\eqno (8.7.3.37)]$ and, by substitution in (8.7.3.34), $[ \mu _{xx}(\rho _{{\rm tot}}) =\mu _{xx} (\Delta\rho) + \textstyle{1\over3}\textstyle\sum \limits _{{\rm atoms}}\langle r^2\rangle _{{\rm spherical\ atom}},\ \eqno (8.7.3.38a) ]$ with $[ \mu _{xx}(\Delta\rho) =\textstyle\sum \limits _{i}\left(\textstyle\int \Delta\rho _{i}x^2{\,{\rm d}}{\bf r}+2X_{i}\mu _{i}+X_{{i}}^2q_{i}\right), \eqno (8.7.3.38b) ]$ in which $[\mu _{i}]$ and $[q_{i}]$ are the atomic dipole moment and the charge on atom i, respectively.

The last term in (8.7.3.38a) can be derived rapidly from analytical expressions for the atomic wavefunctions. Results for Hartree–Fock wavefunctions have been tabulated by Boyd (1977). Since the off-diagonal elements of the second-moment tensor vanish for the spherical atom, the second term in (8.7.3.38a) disappears, and the off-diagonal elements are identical for the total and deformation densities.

The relation between the second moments μ_αβ and the traceless moments $[\Theta]$ _αβ of the deformation density can be illustrated as follows. From (8.7.3.17), we may write $[ \Theta _{\alpha \beta }\left (\Delta\rho \right) = \textstyle{3\over2}\mu _{\alpha \beta }\left (\Delta\rho \right) - {1\over2}\delta _{\alpha \beta }\int \Delta\rho \,r^2{\,{\rm d}}{\bf r}.\eqno (8.7.3.39)]$ Only the spherical density terms contribute to the integral on the right. Assuming for the moment that the spherical deformation is represented by the valence-shell distortion (i.e. neglect of the second monopole in the aspherical atom expansion), we have, with density functions ρ normalized to 1, for each atom $[ (\Delta\rho) _{{\rm spherical}} = \kappa ^3{P}_{{\rm valence}}\rho _{{\rm valence}}(\kappa r) -{P}_{{\rm valence}}^0\rho _{{\rm valence}}(r) \eqno (8.7.3.40)]$ and $[\eqalignno{ \textstyle\int \Delta\rho \,r^2{\,{\rm d}}{\bf r} &=\textstyle\int \textstyle\sum \limits _{i}\left [\kappa _{i}^3{P}_{{\rm valence,}i}\rho _{{\rm valence,}i}\left (\kappa _{i}r\right) \right.\cr &\left. \quad -{P}_{{\rm valence,}i}^0\rho _{{\rm valence,}i}\left (r\right) \right] r^2{\,{\rm d}}{\bf r} \cr &=\textstyle\sum \limits _{i}\left ({P}_{{\rm valence,}i}/\kappa _{{i}}^2-{P}_{{\rm valence}}^0\right) \left \langle r_{i}^2\right \rangle _{{\rm spherical\ valence\ shell}} \cr &\quad+{R}_{i}^2\left ({P}_{{\rm valence,}i}-{P}_{{\rm valence,}i}^0\right), & (8.7.3.41)}]$ which, on substitution in (8.7.3.39), gives the required relation.

References

Boyd, R. J. (1977). The radial density function for the neutral atoms from helium to xenon. Can. J. Phys. 55, 452–455.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 717